Considerations from CFT and from differential geometry indicate that the integral affine structure on $Y$ does not depend on the choice of the Kähler class of Calabi-Yau metrics. Thus, we obtain a “combinatorial” invariant $(Y,T^{{\bf Z}}_{Y})$ of (maximally degenerating) Calabi-Yau variety over the local field $K={{\bf C}}((q))$. One can argue that in this case there will be a canonical atlas of coordinate charts such that the transition maps belong to the group $SAff(n,{\bf Z}):=SL(n,{\bf Z})\ltimes{\bf Z}^{n}$. The natural question arises whether one can define and calculate it purely algebraically, without the use of transcendental methods and Calabi-Yau metrics. We expect that the answer to this question is positive. In other words there exists a canonical way to associate the data $(Y,T^{{\bf Z}}_{Y})$ with arbitrary smooth projective variety $X,\,c_{1}(T_{X})=0$ having “maximal degeneration” over an arbitrary field $K$ with a discrete valuation.

The conjectural answer (only for the compactification $\overline{Y}$ of $Y$) is the following: let us choose (after an extension of the field $K$) a model with stable reduction. Call an irreducible component $D$ of the special fiber $X_{0}$ essential if the order of pole at $D$ of the global volume element on $X$ is maximal among all components of $X_{0}$. We define topological space $\overline{Y}(X_{0})$ as the Clemens complex spanned by essential divisors (see [LTY]). Roughly speaking, $k$-cells of $\overline{Y}$ correspond to irreducible components of $(k+1)$-fold intersections of essential divisors. Recently one of us (M.K.) proved, using ideas from motivic integration and from Berkovich theory of non-archimedean analytic spaces (see [Be]), that for different choices of models with stable reduction spaces $\overline{Y}(X_{0})$ can be canonically identified . In examples coming from toric geometry the space $\overline{Y}=\overline{Y}(X_{0})$ is always a manifold.

It is not clear yet what is the origin of the smooth part $Y\subset\overline{Y}$, and of the affine structure on it. Conjecturally, all this comes from a map $\pi:X(\overline{K})\to\overline{Y}$ where $\overline{K}$ is the algebraic closure of $K$. In the differential-geometric picture of torus fibrations (when $K={\bf C}_{mer}$) the map $\pi$ is obvious: it associates with a meromorphic (finitely ramified) family of points $x_{q}\in X_{q}$ the limit point $lim_{q\to 0}\,x_{q}\in Y$ in the metric sense. Also, the differential-geometric picture suggests that the closure of the image $\pi(Z(\overline{K}))$ where $Z\subset K$ is an algebraic subvariety, should be a piecewise linear closed subset of $Y$, and linear pieces of it have rational directions. In particular, if $Z$ is a curve then $\pi(Z(\overline{K}))$ is a graph in $Y$. This opens a way to express Gromov-Witten invariants of $X$ in terms of the Feynman expansion for certain quantum field theory on $Y$.

Also, we expect that the choice of an ample class in $NS(X)\otimes{\bf R}$ on $X$ gives rise to the dual integral affine structure on $Y$ defined again in some purely algebro-geometric way. If the ample class is the first Chern class of a line bundle, then there should be also a canonical reduction of the dual integral affine structure to a $SAff(n,{\bf Z})$ -structure.